### RESEARCH EXPOSITORY

### AND

### SURVEY

### ARTICLE

### SOME

### FIXED

### POINT ITERATION PROCEDURES

B.E. RHOADESDepartmentofMathematics IndianaUniversity Bloomington, IN 47405

### U.S.A.

(Received August 15, 1989)

### ABSTRACT.

This paperprovides a survey of iteration procedures that have been used toobtainfixedpoints for maps satisfying a variety ofcontractive conditions. The author does not claim toprovide complete coverageofthe literature, and admits to certain biases in the

theoremsthat arecited herein. Inspite oftheseshortcomings, however, this papershould be

auseful reference for those personsvishingtobecome better acquaintedwiththearea.

### KEY

WORDS AND PHRASES.fixedpointiteration.1980 AMS

### SUBJECT CLASSIFICATION

CODE. 47H10, 54H251.

### ITERATION

### PROCEDURES.

The literature abounds with papers which establish fixed points for maps satisfying a

variety ofcontractive conditions. Inmostcasesthe contractive definition isstrongenough,not onlytoguaranteethe existence ofauniquefixedpoint, but alsoto obtainthatfixedpoint by repeatediteration of thefunction.

### However,

forcertain kinds of maps, suchas nonexpansivemaps, repeatedfunction iterationneed not converge to a fixed point.

### A

nonexpansive map satisfiesthecondition### IITx

### Tyll <_

### ]Ix

### Yll

foreach pair of points x, yinthespace.### A

simple exampleisthefollowing. Define### T(x)

1-xfor0### _<

x### _<

1. Then### T

isanonexpansive selfmap of### [0,1]

withauniquefixedpointatx### 1/2,

but,ifonechooses as astarting point the valuez a,a

### 1/2,

then repeatediterationofTyields thesequence### {1

-a,a, 1### -a,a,...).

### In

1953### W.R.

Mann### [32]

defined the following iteration procedure. Let### A

be a lower triangularmatrix withnonnegativeentriesandrowsums 1._{Define xn+l}

_{T(vr,),}

where
Vn

### E

ankXk"k=O

The mostinterestingcasesof theManniterativeprocessareobtainedby choosingmatrices

### A

such that### a,+l,k

### (1-a,+,,+)a,t,,k

0,1,...,n;n 0,1,2,...,andeither a,, 1 for alln ora,,### <

1for alln### >

0. Thus,ifonechoosesany sequence### {c,}

satisfying### (i) co

1,### (ii)

0

### _<

c,,### <

1 forn### >

0, and### (iii)

c, c, thenthe entriesof### A

becomeatn C/t

ank=ck

### H

### (1-cjl,

k<n.### (1.1)

j=k+l

2 B.E. RHOADES

Theabove representation for

### A

allowsoneto writetheiterationschemein thefollowing form: x,+,### (1

### c,.,)x,.,

### +

c,.,### T(

x,### ).

One exampleofsuchmatrices is the

### Cesaro

matrix,obtainedby choosingc,,### 1/(n

### +

### 1).

Anotheris

_{c.}

1 for alln ### >_

O, which correspondsto ordinaryfunctioniteration, commonly calledPicard iteration.Pichard iterationof the function

_{$1/2}

### (I

### +

### T)/2,

is equivalent to theManniterationschemewith

ank

-This matrix isthe Eulermatrixof order 1,and the transformation

_{SI/}

has beeninvestigated
byt’]delstein_{[13]}

and Krasnoselskii### [30].

Krasnoselskii showed### that,

if### X

isauniformlyconvexBanach space, and

### T

is a nonexpansive selfmapof### X,

then_{5’/}

converges to a fixed point
ofT. Edelsteinshowed that theconditionofuniformconvexity could be weakenedtothatof
strictconvexity.

Pichard iteration of the function

### Sx

### AI

### +

### (1

### A)T,

0### <

### <

1, for any function### T,

homogeneousofdegree 1,isequivalent to theManniteration schemewith

This matrix isthe Eulermatrixoforder

### (1

### A)/A.

Theiterationof_{5’x}

has beeninvestigated
by Browder andPetryshrn ### [7],

Opial### [36],

and Schaefer### [50].

Mannshowed

### that,

if### T

is any continuousselfmapofa closedinterval### [a,

### b]

withatmostonefixed point, thenhis iterationscheme,with c,,

### 1/(n

### +

### 1),

converges to thefixedpoint of T. Franks and### Marzec

### [15]

extendedthisresult to continuous functions possessingmorethanonefixedpointintheinterval.

### A

matrix### A

iscalledaweightedmeanmatrixif### A

isalowertriangularmatrix with nonzero entries### an

### p/Pn,

where### {p

### }

isanonnegative sequence withp0 positiveand### Pn

### Z

p cx,k--0

The author

### [42]

extended the above-mentioned result of Franks andMarzec toanycon-tinuousselfmapofaninterval

### [a, b],

and### A

anyweightedmeanmatrixsatisfying the conditionlim

### ]an,

an-l,t O.k--0

### In

_{[42]}

the author also showed that thematrix definedby### (1.1)

isequivalent toaregular weightedmeanmatrix withweightsckp0

k

### >

0.### H(

### )’

### Let E

be aBanach space,### C

aclosedconvexsubset of### E,

### T

acontinuous selfmap of### C.

Mann

### [32]

showed### that,

if either of the sequences_{{xn}}

or ### {vn }

converges, then sodoes the### other,

and to thesamelimit, whichis afixedpoint of### T. Dotson

### [12],

extendedthisresult to locallyconvexHausdorfflineartopologicalspacesE. Consequently,tousethe### Mann

iterativeForuniformlyconvexBanachspaces,the followingwasobtainedindependently byBrowder

### [6],

Kirk### [281,

and Gohde### [6].

Let C beaclosed,

### bounded,

andconvexsubset ofauniformlyconvex Banachspace,### T

anonexpansiveselfmapofC. Then

### T

hasafixedpoint.Unfortunatelythe proofs of above theoremarenotconstructive.

### A

number of papers have beenwritten to obtainsomekindof sequential convergence for nonexpansivemaps. Mostsuch theoremsarevalidonlyundersomeadditional hypothesis, suchascompactness,andconverge only weakly. Someexamples oftheorems ofthis type appear in### [8].

Halpern### [18]

obtained two algorithmsfor obtaining fixed points for nonexpansive maps on Hilbert spaces.### In

hisdissertation, Humphreys

### [23]

constructed an algorithm which canbeapplied toobtain fixed points for nonexpansivemapsonuniformlyconvexBanachspaces.### A

nonexpansivemappingis saidto be asymptotically regularif, for each point x inthe space,### lim(T"+lx

### T"z)

0.### In

1966 Browder and Petryshyn### [7]

established the following result.THEOREMI.

### ([7,

Theorem_{6]).}

Let### X

beaBanachspace,### T

anonexpansive asymptot-icallyregularselfmap ofX.### Suppose

that### T

hasafixedpoint, and that### I- T

mapsbounded closed subsets of### X

intoclosed subsets ofX. Then,for eachz0 E### X, {Tnx0

converges toafixedpoint of

### T

inX.### In

1972 Groetsch### [17]

established the following theorem, which removes the hypothesis that### T

beasymptoticallyregular.### THEOREM

2.### ([17,

Corollary### 3]).

### Suppose T

isanonexpansive selfmap ofaclosedconvexsubset

### E

of### X

whichhas at leastonefixed point. If### I-

### T

mapsbounded closedsubsets of### E

intoclosed subsets of### E,

then the Manniterativeprocedure, with### {c,}

satisfyingconditions### (i), (ii),

and_{(iv)}

_{c,,(1}

_{c,,)}

oo, convergesstronglytoafixedpoint ofT.
Ishikawa

### [25]

established the following theorem.### THEOREM

3. Let### D

beaclosed subset ofaBanachspace### X

and let### T

beanonexpansivemapfrom

### D

intoacompact subsetofX. Then### T

hasafixed pointin### D

and theManniterative process with### {c,}

satisfyingconditions### (i)- (iii),

and 0### _<

c,### <

b### <

1for all n, convergestoafixedpoint ofT.

For spaces ofdimension higher than one, continuity is not adequate to guarantee

con-vergence toafixed point, either by repeated function iteration, or by some otheriteration

procedure. Therefore it is necessarytoimposesomekindofgrowthconditionon themap. If thecontractivecondition isstrong enough,then themapwillhaveaunique fixed point,which

canbeobtainedby repeatediterationof thefunction. If thecontractive condition isslightly weaker, thensomeotheriterationschemeisrequired. Evenif the fixed pointcanbe obtainedby functioniteration,itisnotwithoutinterestto determine ifotheriteration

### procedures

converge tothefixedpoint.### A

generalizationofanonexpansive map with at leastonefixed point that ofa quasi-nonexpansive map.### A

function### T

isaquasi-nonexpansivemapifithas at leastonefixed point,### and,

for eachfixedpointp,### [[Tx

### -p[[

### _<

### [Ix

### -p[[.

The followingisdue toDotson### [12].

### THEOREM

4. Let### E

beastrictlyconvexBanach space,### C

aclosedconvexsubset of### E,

### T

acontinuousquasi-nonexpansiveselfmapofCsuch that

### T(C)

C### K

C### C,

where### K

iscompact. Let x0 E### C

and consider a### Mann

iterationprocess such that### {c,}

clusters at somepoint in### (0,1).

Then the sequences### {x,,},

### {v,}

convergestronglytoafixed point of### T.

4 B.E. RHOADES

thespace, atleast oneof the followingistrue:

### (i)

_{IlTx}

### Tyll

### <_ ,llx

### yll, (ii)

### []Tx

### Ty[]

### <_/3[[[x

### Tx[[

### +

### ]]y- Ty[[],

or### (iii)

### [[Tx

### Ty[[

### <_

### 3’[[Ix

### Ty[[

_{+}

### [[y-

### Tx[[],

where a,/3,3’ arereal nonnegative constantssatisfying a

### <

1,/3,3’### <

### 1/2.

Asshownin### [52],

### T

has a unique fixedpoint, which can be obtainedby repeatediteration of the function. The following resultappears in

### [42].

THEOREM5.

### ([42,

Theorem### 4]).

Let### X

beauniformlyconvexBanachspace,Eaclosedconvexsubset of

### X, T

aselfmapof### E

satisfyingconditionZ. Then the### Mann

iterativeprocesswith

### {c,}

satisfyingconditions### (i),

### (ii),

and### (iv)

convergestothe fixedpoint ofT.### A

generalization of definition### Z

was made by Ciric### [11].

### A

mapsatisfies condition Cifthereexists a constant k satisfying 0

### _<

k### <

1 such### that,

for eachpair of points x,y inthespace,

### IlTx

### Tyll

### <

k_{max{llx}

### ll,

_{Ilx}

### TxII,

### Ilu

### TyII,

_{II}

### Tull,

### II

### In

### [42]

theauthorproved thefollowing for Hilbertspaces.THEOREM6.

### ([42,

Theorem### 7]).

Let### H

beaHilbert space, Taselfmapof### H

satisfyingcondition C. Then the Mann iterative process, with

### {c,}

satisfying conditions### (i)-(iii)

andlimsupc,

### <

1 k2_{converges}

_{to}

_{the fixed point of}

_{T.}

Chidume

### [10]

has extended the above result to gv spaces, p### >

2, under the conditions### k2(p-1)<

1 andlimsupc,### <(p-1)--k

### .

Asnoted earlier,if

### T

iscontinuous,then, iftheManniterativeprocess converges,itmust converge toafixedpoint of### T.

If### T

is not continuous, thereis no### guarantee

that,evenif the### Mann

processconverges,itwill converge toafixed point of### T.

Consider,forexample,themap### T

definedbyTO T1 0,### Tz

1,0### <

z### <

1. Then### T

isaselfmap of### [0,1],

withafixedpointat x 0.

### However,

theManniterationscheme,with_{cn}

_{1/(n}

_{+}

### 1),

0### <

0### <

1, convergesto1,which isnot afixed point ofT.

### A

Inap Tis said tobe strictly-pseudocontractiveifthere existsaconstant k, 0### _<

k### <

1 suchthat,for all pointsx,y in thespace,### [ITx

### Tyll

### <_

_{Ilx}

### y]l

_{+}

### kll(I

### T)z -(I

### T)yII

### .

Weshall call denote the class ofall suchmapsby

### P2.

Clearly### P2

mappings contain thenonex-pansivemappings,but the classesP2,

### C,

andquasi-nonexpansivemappingsare independent. THEOREM 7.### ([42,

Theorem### 8]).

Let### H

beaHilbert space,### E

acompact,convexsubset of### H,

### T

a_{P2}

selfmapofE. Then the Mann iteration scheme with### {c, }

satisfying conditions### (i)-(iii)

and limsupc, c### <

1 k,convergesstrongly toafixedpoint ofT.### A

pseudocontractive mappingis a_{P2}

mappingwith k 1. ### Let

### P3

denotethe familyof pseudocontractivemappings. Hicks andKubicek### [22]

gave anexaxaple ofa_{P3}

mappingwith
afixedpoint such that the

### Mann

iterativeprocedure, withc,.,### 1/(n

### +

### 1),

doesnotconverge tothefixedpoint.Ishikawa

### [24]

defined the followingiterativeprocedure. Given anyx0 in the space, definex,+,

### a,T[nTx,

### +

### (I

### ft,)x,.,]

### +

### (I

where

_{{a, }, {/3, }}

aresequences of positive numberssatisfyingthe conditions 0### <

c,, _</3.### <_

1,lim/3. 0, c,,13. oo.

### He

thenestablished thefollowing result.THEOREM8. Let

### E

beaconvex, compactsubset ofaHilbert space### H,

### T

alipschitzianQihou

### [38]

extended the above result tolipschitzian hemicontractive maps.### A

hemicon-tractive map is a pseudocontractive map withrespect to afixed point; i.e., if p is any fixedpoint of

### T,

andx is anypointinthespace,then### T

satisfies_{IITx}

### pll

### <_

_{IIx}

### p]l

### +

### ]Ix

### Tx]]

### .

Neitherthe proof of Qihounor Ishikawacanbe used toestablishasimilarresult for the Manniterativetechnique.### In

1968Kannan### [27]

introducedacontractivedefinition which did not require continuity of themap. There then followedaspate offixedpoint papers dealingwithfixed points foravariety of mapswhich werenot necessarilycontinuous. For example, maps oftype C and

### Z

need not becontinuous. Inanearlier paperthe author### [45]

partially ordered manyof these definitions.### As

noted above,it has been shown that theMann iterationscheme converges forsomeofthese. Forothercontractive definitionsit isnotpossible todeterminewhetherornot the

### Mann

orIshikawaiteration processesconverge.### However,

in some eases it is possible to showthat, iftheydo converge, thentheymust convergetoafixed point ofT. The following theorems illustratethis idea.### THEOREM

9.### ([42,

Theorems### 5,6]).

### Let

### X

beaBanachspace,### T

aselfmapof### X,

### T

EC or### T

E### P2.

Let### {c,,}

satisfy theconditions### (i), (ii),

andboundedawayfromzero. Then,ifthe corresponding Manniteration processconverges,itconvergestoafixedpointofT.### THEOREM

10.### ([46,

Theorem### 1]).

Let### X

beaclosedconvexsubset ofanormed space,### T

aselfmapof### X

satisfyingthepropertythat there exist constants c, k### >

0,0### <

k### <

1 such that,for each x,y### X,

### IITx

### Ty[[

### <_

k_{max{cllx}

_{ttll, [ll}

_{Txll}

/### I[Y

### TYH],

_{Ill}

x ### TyII

_{+}

### Ily

### TII]}.

### Let

### {c,,}

satisfyconditions_{(i), (ii),}

and limc, ### >

0. If theMann iterationscheme converges, thenit convergestoafixedpoint ofT.### A

contractivedefinitionwhich isindependent of that of Ciric isthe following, duetoPal andMaiti_{[37].}

### For

each pair of pointsinthespace,atleastoneof thefollowingconditionsis satisfied:### (i)

### [Ix

### Tx[[

### +

### [[y

### Ty[[ <_

### a[[x

### y[[,

1### <

ot### <

2,### (ii)

### [Ix

### Tx[I

### +

### I[Y- TYI[

_{<-}

### {[[x

### Ty]l

### +

### I[Y- Tzl[

### +

x### Y]I},

### 1/2

_</3### < 2/3,

### (iii)

### ]Ix

### Tx]]

### +

### I]Y

### Ty]]

### +

### ]]Tx

### Ty]]

### <_

### 7{]Ix

### Ty]]

_{+}

### ]]y

### Tx]]},

1### _<

7### <

### 3/2,

### (iv)

_{IlTx}

### Tyll

### <_

6### max{l]x

### Y]I,

### [[x

### Tx][,

### ]IY

### Ty]],

### []lx

### Tyll

_{+}

### [[y

### Txl[]/2

### },

0### <

6### <

1.Using this demitiontheauthor established thefollowingresult.

### TI-ImOIM

ll.### ([46,

Theorem### 2]).

Let### X

beamanaeh space, aselfmapof### X

satisfying the bove contractive definition. Then if the### Mann

iteration### scheme,

with### {cn}

satisfying conditions### (i), (ii),

andlim### ,

### >

0, converges,itconvergestoafixed point of### T.

Similar results have been established for the Ishikawa procedure.

### In

its original form the Ishikawaprocedure does notincludetheMann iteration processbecauseof the condition 0### <_

### cn

_</3n### _<

1.### For,

ifthe Ishikawaprocesswere toincludetheMann processas aspecial case, thenonewould have to assign### each/3n

tobe zero, forcing then each### cn

tobezero.### In

aneffort to haveanIshikwatypeiterationschemewhichdoesinclude theManniterativeprocess

asaspecial case,someauthorshavemodifiedtheinequalityconditiontoreal 0

### _<

### n,/3n

### _<

1. This### change

isreflectedinthefollowingtwotheorems.### THFOREM

12.### ([35,

Theorem_{1.2]).}

### Let

### X

be anormed linearspace, (7aclosedconvex6 B.E. RHOADES

If the Ishikawa iteration scheme, with the

### an

bounded away from zero, converges to apoint p,thenp isafixed point ofT.

THEOREM 13.

### ([40).

Let### E

beaclosedconvexsubset ofaBanachspace,### X, T

aselfmapof

### X

satisfying the condition: thereexists constantsc,k### _>

0,0### _<

k### <

1, such that, for each pair of points z,y in### X,

### ][Tx

### Tyl[

k### max{cllx

### ll,

### [}lx

### Txll

/### Ily

### TYl]],[]I

x### TyII

/### Ily-

### Txll]).

If theIshikawascheme, with

### {an}, {fin}

satisfying theconditions 0### <

a### _<

a,,### _<

1, 0### _<

fl,### _<

1and limfl, 0, converges toapointp, thenp isafixed point ofT.

### In

some theorems ofthistype,the contractivedefinition is weakenoughthat theaprioriexistence ofa fixed point for

### T

cannot be assured. Thus the convergence of the iteration procedure also yields the existenceofafixed point.The literature also contains theorems ofthis type for other contractive definitions. In

1988 the author

### [47]

proved### that,

formany contractive definitions, eventhoughthe function need not becontinuouseverywhere,it isalwayscontinuousinaneighborhoodofafixed point. Therefore,assuming the convergence of theMann procedureistantamount,insomecases, toassuming theconvergencetoafixedpoint,inlight of the early result ofMann

### [32].

Wenowreturntononexpansivemaps.

THEOREM 14.

### ([1]).

Let### K

beaclosedboundedconvexsubset ofaHilbertspace### H,

### T

a nonexpansiveselfmapofK. Foreach pointeof### K,

theCesarotransform of### {T’e}

converges weaklytoafixed point of### T.

If

### T

is also an odd map, then Baillon### [2]

showed that the convergence of the Cesaro transform of### {The}

isstrong. Healsoextendedthe above theorm to### L

_{’}

spaces.
Wenowmentionsomeresults for othermatrixtransformsof iteratesofT.

THEOREM 15.

### ([5,

Theorem### 1]).

Let### T

beaHilbert space, Caclosed boundedconvexsubset of

### H,

### T

anonexpansiveselfmapof### C.

### Suppose

that### A

isan infinite matrixwith zerocolumn limits and satisfies lim,,

### k(a,,,k+l

### a,,k)

### +

0. Then, for eachx in### C,

### {

### a,.,,T’x}

converges weakly toafixedpoint ofT.

### A

sequence x is said to be almost convergent to a limit q if### lim,[x,,

### +

x,,+l### +...

### +

x,.,+,_

### ]/n

q, uniformly inp.### An

infinite matrix### A

issaid to bestrongly regularif### A

assignsa limit toeach almost convergent sequence.

### Necessary

and sufficient conditions for### A

to be strongly regularwereestablishedbyLorentz### [31].

Theseconditionsarethat### A

be### regular

and alsosatisfy lim,_{E}

### la-,*+

### a-l

0.### A

uniformlyconvexBanachspace### X

hasamodulus ofconvexity### $(e)

definedby### (5(e)

### ink’{1

### ]Ix

### yll/2

_{IIll}

1,_{IIll}

1,_{I1}

### yll

### },

0### <

2.Let

### {u,,}

beabounded sequence inaclosedconvexsubset CofX. Define### rm(x)

### sup{llu.

### zll-

n### >_ m},

d denote by c,,, the unique point in C with the property that### r,,,(c,,,)

### inf{rm(x)’x

E### C}.

Then### limcm

c, andcis called the asymptotic center of### {u,,}.

### (See,

e.g.### [41)

Bruckprovedthefollowingtheorem.

### C, A

astrongly regularmatrix, the A-transform of### T"x

convergesweaklytoafixed pointp, which is the asymptoticcentcr of### {T’x}.

Schoenberg

### [51]

obtainedthefollowingcharacterizationfor nonexpansive mapson aHilbert space.### THEOREM

17.### ([51]).

Let Tbeanonexpansive ofaHilbertspace### H,A

beanonnegativematrixwithrowsums oneandzerocolumn limits,

k----0

Then thefollowingareequivalent;

### (a) {S,}

isweakly convergent tothe asymptotic centerzof### {T"x}.

### (b) {,.,c

isweakly convergent toafixed point for### T.

### (c)

If_{V}isaweak limit point of

### {S,},

then_{lim,...oo(Re(T"z}

### T"+*z,z

### Ty))

0### (d)

Ifyisaweak limit point of### {S,},

then### (e)

Each weak limit point of### {S,}

is afixedpoint ofT.Thefollowingtheorem establishesafixed point foracontractive definition which includes

nonexpansivemappings.

### THEOREM

18.### [34].

Let### X

be auniformly convex space,### K

a nonempty closed and uniformlyconvexsubset of### X,

### T

aselfmapof### K

satisfying### IITx

### Tyl[

### <

### a(x,

### y)llx

_{Yll +}

### b(x,

### y)llx

### b’(x,y)lly

### Tyll

### +

### c(x,y)llx

### Tyll

_{+}

### c’(x,y)lly

_{Txll}

### (1.2)

where### a,b,b’,c’

### >

### O,(a

### +

### b+

b’_{+}

c### +

### c’)(x,y)

### <

1, and### b’(z,y)

### b(y,z),c’(x,y)

### c(x,y),

for allx,yin### X.

Ifsup

### (b+c’)(x,y)

### <

1### (1.3)

and

inf

### IIx

_{TxlI}

0 forsomefiniter
then### T

hasafixedpoint in### K.

Define

### (1.4)

### (y)-

_{a.lly-}

### T# ell",

n O, 1,2,...,1### <

p<co.### (1.5)

j=0where

### A

isanynormegative triangularmatrixwithrow sumsone, y, e, E### K. For

eachndefine### s

tobethe unique point of### K

where### (1.5)

assumes its minimum.### THEOREM

19.### ([48,

Theorem### 2]).

### Let

1### <

p### <

oo,### X

### e’,K

a nonempty closed,8 B. IS. RItOADES

THEOREM 20.

### ([48,

Theorem_{3]).}

Let 1 ### <

p### < o,X

### gp,

### K

a noneinpty closed, bounded, convex subset of### X,

T a quasi-nonexpansive selfmap of Ix’. If each subsequential weaklimit of### {s

isafixedpoint of### T,

then### {s

converges weakly toafixedpoint ofT.The specialcaseoftheabovetheoremsin which

### A

is the Cesaromatrix and### T

isnonex-pansive appear in

### Beauzamy

andEnflo_{[4].}

Weshallnowexaminetwootheriterationmethods forobtainingfixedpoints. Thefirstof theseis dueto Kirk

### [29]

and is defined asfollows. Let### {x,,}

be any sequence of pointsinthespace. Then theiterationprocedureisdefinedbythe operator

S

### a,T’,

### (1.6)

’-0

where kisafixedinteger, k

### >_

1,a,### >_

0, 0,1,2,...,a### >

0 and### =0

a, 1.For

### T

any nonexpansive selfmap ofa convexset, S and### T

have the samefixed points.### (See

### [29].)

LetTbea selfmapofa Banachspace

### X

satisfying, for each x,y in### X,

### IIT:

### Tull

### <_

### mx{ll:

### ulI,

### [11:

### T:II

### + Ilu

### TulI]/.,

### [11:

### Tull + Ilu

### T:II]/2}.

### (1.7)

In

### [41]

it is shown that, ifS satisfies### (1.6),

then S and### T

have the samefixed points. Thefollowing result is then proved.

THEOREM 21.

### ([41,

Theorem### 2]).

Let### X

be a uniformly convex Banach space,### K

abounded closedconvexsubset of

### X,

Taselfmap of### K

which satisfies condition### (7),

Sdefinedby

### (6).

If### I-

Smapsbounded closed subsets of### K

intoclosedsets, then,for each x0 E### K

thesequence

### {S"x0 }

convergestoafLxed point of### T

in K.Theotherfixed point iterationprocedureisdefinedby

### s=

a, 1,### aa+

7 0 for at least oneinteger k.=0

Massa

### [33]

has shovnthat,ifTisquasi-nonexpansive, then Sand### T

have thesamefixedpoints. Hehas also provedthefollowing.

THEOREM22.

### ([33,

Theorem### 2]).

LetIfbeaclosedconvexsubset ofauniformlyconvexBanachspace

### X,

### T

aselfmap of### K

satisfying### IITx

### Tull

### <_ ,(, u)llx

### Yll

### +

### b(:,

### g)(ll:

_{T:II}

### + Ilu

### TulI)

### +

### ,:(:,

### Y)(II

### Tull

### +

### Ilu

### T:II),

where a,### b,

c,### >_

0,### (a

### +

### 2b+

### 2c)(x,

y)### _<

1,inf,yeK### b(x, V)

### >

0,and### T

has atleastonefixed point. Then_{{S"x}

converges to theonlyfixedpoint ofT.
2.

### RATE OF CONVERGENCE.

There hasbeennosystematicstudy of therateof convergencefor these iteration

### proce-dures,

andit isdoubtfulif anyglobalstatementscanbemade,sincethereisnothing about theseiteration proceduresto cause theiranalysis to bedifferent fromthat of other approximation methods.

Theauthor

### [44]

obtainedsomeevidenceonthebehaviorof the### Mann

iterationprocedure for the decreasing functions### f(x)

1### xrn,g(x)

### (1

### z)

for 1### _<

rn <: 6 and c,,bisectionmethod,accurate to 10places. Then both theManniterationand Newton-Raphson methodswere employed to find each fixed point to within 8 places, using initial guesses of

x0 .1,.2,...,.9. The author

### [43]

also examined thefunctions gform 7, 8,...,29, withaninitial guess ofx0 .9,anda,

### ,

### (n

### +

### 1)

### -1/2. In

eachcasetheManniterativeprocedure convergedto afixedpoint### (accurate

toeight### places)

in9-12iterations, whereas the Ishikawamethod required from 38 to 42 iterationsfor the same degree ofaccuracy. For increasing

functionsthe Ishikawamethodisbetter than theMannprocess, but ordinary functioniteration isthebest of all three for increasing functions.

### An

examinationof the printout showed that theNewton-Raphsonmethod converges faster than the### Mann

scheme. This isnot surprising,since theNewton-Raphsonmethod converges quadratically, whereas the Mann process converges linearly.### However,

whereas the rate of convergenceofthe Newton-Raphson methodis very sensitive to thestarting point,the rate ofconvergence for theMannprocessappears to be independent of theinitialguess.

3.

### STABILITY.

Weshallnowdiscussthe question of stability ofiteration processes,adopting thedefinition

of stability thatappears in

### [19].

Let

### X

beaBanach space,### T

aselfmapof### X,

andassumethat_{xn+l}

### f(T,x,)

definessomeiterationprocedure involving T. For example,

### f(T, Xn)

### Tz

### Suppose

that### {

x, converges toafixed pointpofT. Let

### {y,}be

anarbitrarysequence in### X

anddefineforn 0,1,2, Iflim, 0 implies that lim, y, p, then theiteration procedure

### f(T,

### x)

issaidto beT-stable. ThefirstresultonT-stable mappingswasprovedbyOstrovskifor the Banachcontractionprinciple.

### In

### [20]

the authors show that functioniteration isstable foravariety ofcontractivedefinitions. Their best result forfunction iteration isthefollowing.### THEOREM

23.### ([20,

Theorem### 2]).

Let### X

be a complete metric space,### T

a selfmap of### X

satisfying thecontractive condition ofZamfirescu. Let p be the fixed point ofT. Let### xo

### _

### K,

set x,+l### Txn,n >_

O. Let### {y,}

beasequence in### X,

and set### ,

### d(y,+l,Ty,)

forn 0,1, 2, Then

### <

_{+}

_{+}

0,
k=O k-O

where

=max

### a,l_,l_7

and

lim y, pif andonlyif lim e

### n=0.

### For

theManniterationproceduretheirbestresultis thefollowing.### THEOREM

24.### ([20,

Theorem### 3]).

Le### (X,

### [[.[[)

beanormed linear space,### T

aselfmapof### X

satisfyingthe Zamfirescu condition.

### Let

x0 E### X,

and suppose that thereexistsafixed pointpand z, p, where

### {x,}

denotes the### Mann

iterativeprocedureswiththe### {c,}

satisfying### (i),

### (ii),

and0

### <

a### <

### c.

### <

b### <

1.### Suppose {y.}

isasequencein### X

### ande.

### IIv,.+-[(1-c.)y.+c.Tv.]II

0 B.E. RHOADES

### liP-

### Y-+I]

_{-<}

_{liP-}

### x-+ll +

### (1

a4-### a)n+l

### [Ix0

### 2b(1

12-[-### a)n+l

_{[Ix,- Tx,[[}

### +

### (1

a### +

=0 =0

where

### {

t=max

_{C,l_fl,}

l_7
and

n=0,1,2,...,

lim y,, pif andonlyif lim

### ,,

0.Fortheiterationmethod of Kirk

### [29],

they have the following result.THEOREM 25.

_{([20,}

Theorem ### 4]).

Let### (X,

### I1"

### II)

be Ba=chspce,### T

seifmp of### X

satisfyingtheBanachcontractivecondition

_{ilT-Tull}

### <

_{cllx-vll}

forsomeconstant c,0 ### <_

c### <

1.Letp bethe fixedpoint of

### T,

z0 an arbitrary point ofX. Setk

Xn+l

### a,T’x,,

forn 0, 1,2,...,t--0

wherekisaninteger such that k

### >_

1,a,### >_

0for 0,1, 2,...,k,al### >

0,and ke,,

_{l[,}

_{aiT’Y’ll}

forn 0,1,2,....
t-’0

Then

i=0 t=O j---0

n 0,1,2,...

and

lim y,, pif and onlyif lim,, 0.

### In

arecent paperthe author### [49]

has### proved

each of the above theorems foracontractivedefinitionindependent of that ofZamfirescu.

Consider thefollowingcontractivecondition: thereexistsaconstantcsatisfying0

### <

c### <

1suchthat,for each pair ofpointsx,y in

### X,

### IITx

### Tyll

### <

### cmax{llx

### Vii,

### Ilx

### TVlI,

### IlU

### Txll}.

### (3.1)

Ourfirstresultisthe following.

THEOREM 26.

### ([49,

Theorem### 1]).

### Let

### (X, d)

bea completemetricspace,### T

aselfmap of### X

satisfying### (3.1).

Let p be the fixed point ofT.### Let

x0 E### X

and definex,,+l### Tx,. Let

### {y,,

### }

CX. Define e,-,### d(y,,+

l,### Ty,,).

Then### d(p,

yn+### <

### d(p,

Xn+l### +

### rrlcn+l-k

### d(xk,

Xk+ -1-### crt+l

### d(xo,

Yo### "+

### Crt-k

k,where m

### 1/(1

### c),

and lim,-oo pif andonlyif lim,,_o### ,

0.### THEOREM

27.### ([49,

Theorem### 2]).

Let### (X,

_{I1"}

### II)

be a Banachspace,### T

aselfmap of### X

satisfying

### (3.1),

pthefixedpoint ofT. Let### {z,}

denote theMann iterativeprocess withthe### {c,}

satisfying### (i)- (iii),

lim

### II(1

ci### +

cci### o,

and### [xx

### (1

c,### +

cc, converges .=0i=j+!### Let (y,}

C### X

and define### ,,

### Ily.+

### (1

### c.)y.

c,### Ty,

_{ll.}

Then
### lip-

./,### <

lip-i=0i=j+l

### H(1

c,### +

### ,:,)11o

### uoll +

### (1

c,### +

### cc,)e.,

i=o /=oi=j+

whereit isunderstood that theproductis1 when j n. Then

hm /,,=p ifandonlyif lim e,=0.

Thestabilityresult forKirk iterationshas been extended to thetwoindependent contrac-tivedefinitionsmentioned earlier.

### THEOREM

28.### ([49,

Theorem### 3]).

Let### X

beaBanachspace,### T

aselfmap of### X

satisfyingcondition

_{(3.1).}

Let p be the fixed point of### T,

### xo

E### X,

and define### {Z,+l}

as in### (1.6).

Let### {!/,,

### }

C### X,

anddefinek

### .

### Ilu.

### ,T’u.II

fo. 0, 1,2,### Then,

withm_{1/(1- c),}

j=0 i=0 i=0

### ._

_{(EiCi)n+l}

_{I1 o}

_{ o11/}

_{(o, 0}

i=0 j=0 i=0

and lim,,_.o. p if andonlyif lim,-ooe, 0.

### THEOREM

29.### ([49,

Theorem### 4]).

Let### X

beaBanach space,### T

aselfmapof### X

satisfyingconditionZ.

### Let

p be thefixedpointofT,z0E### X,

and define### {Z,+l}

asin### (1.6).

### Let

### {/,}

C### X,

anddefine

k

### ,,

_{IIv.}

### ,T’u.II

fo., 0, 1, 2,....12 B.E. RHOADES

Then, with

### {

=max a,

### 1_,1_

7### +

t=0 ./=0

vand lim,-oo y, p if andonlyiflim,-.ooe, 0.

Weshallnowprove that theIshikawa iterationprocedure is T-stablefor mapssatisfying

either condition

### Z

or### (3.1).

THEOREM 30. Let

_{(X,}

_{I1"}

_{II)}

be aBanachspace,### T

aselfmap of### X

satisfying### (3.1).

Let### {x, }

bedefinedbytheIshikawaprocess; i.e.,Xn+

### (1 -a.)x.

### +a.Tz.,z.

### (1

### .)x,,

### +

### .Tx.,

where 0</,,### <

1,0### <

a### <

or.### <

1for all n, and the

### {a.}

satisfy### lim.

### 1-Ii"=l(1-

ai### +

### cai)

0 andconverges.

Let

### {y}

C### X,

anddefineThen

r-0

,=0 j=i+l

.--I

### E

### H

### (1-

a.i### +

### ca:i)i,

t=O

wherem

### c/(1

### c),

and### lim.-oo

y. pif andonlyif### lim.-oo

### .

0.Proof. Letpbe thefixedpoint ofT. Then

### (3.2)

### II.+i-vll-<

### ll(1

### a.)x, +a.Tz,,-vll _< (I-

### .)II,,-vll

### +

### a.llTz.-pll.

### IIT,,

### Tpll _<

### m{ll,,

### r’ll,

### II,,

### TPlI,

_{II’-}

### T.II)

### II,,

### PlI.

### I1=.

### pll

### I1(1

### .).

### +

$.Tz.### pll

_{-<}

### (1

### .)llz.

### pll

### +

### .IIT.

### pll

### < (1

### /.

_{+}

### c/.)11.

### pll

### -<

_{I1.}

### pll.

Therefore

### [[z.+,

### PI[

_{-<}

### (1

### a.

### +

### ca.)[Ix.

### p[[,

and### {[Izn

### p[[}

ismonotone decreasing inn, and soconverges to a limit d

### >

0. If d### >

0, then, taking the limit as n oo, in the above inequality, yields d### _<

_{(1 -a(1 -c))d <}

d, acontradiction. Therefore d 0 and ### {z.}

converges top. Using

### (3.1),

### [Ix.

### Tx.[[ <

### [Ix.

### p[[

_{+}

### [[Tp

### Tx.[[

### < (1

_{+}

### c)[[x.

### p[[,

and### lim.

### [Ix.

### Tx.[[

0.### Note

alsothat_{x.}

### -

pimplies### z.

p, andhence### [[z.

### Tz.[[

O.### For

any x,yin### X,[[Tx-

### Ty[[

### <_

_{m[[x-Tx[[}

_{+}

### c[[x-

### y[[,

wherem_{c/(1}

### -c).

Thus### Ilu.+,

### pll

### <

_{I1.+}

_{vii}

### +

### I1(1

### .).

### +

a,.,Tz.### (1

### c.)y.

### a.T[(1

### /.)y,,

### +

### Z.Tu.]II

### +

### .

But

SO

and

_{(3.2)}

followsbyinduction.
### An

infinitematrix### A

iscalled multiplicative if theA-limitofaconvergentsequenceisequal tosomeconstantmultipleof thelimit of the sequence. If thematrix haszero columlimitsand finite norm, then the multiplieristhelimit oftherow sums.

Returning to the proof of thetheorem, suppose that

### ,,

0. Let### A

denote the lower triangularmatrixwithnonzeroentries a,. mc,0,a.k ma._k

### 11

### (1

trj### +

### coL/),

k### <

n. j=k+l### Then,

since### Ilz,,

### Tz.[[

### O,

### []z.

### Tz.[[

### O,

d### A

ismultiplicative, therighthdsideof

### (3.2)

tends tozero n### ,

d p.### Suppo

### .

p.0

### e.

### fly.+,

### (I

### a.)y.

### a.T[(l

$.)y._{+}

### D.T.]II

### Uy.+,

### p]]

### +

### (1

### a.)][y.

### p[[

### +

### a.][T[(l

### .)y.

### +

### .ry.]

### rp[]

14 B.E. RHOADES

THEOREM 31. Let

### (X,

_{II}

### ]l)

bea Banachspace,### T

aselfmap of### X

satisfying conditionZ. Let

### {

z,### },

y,### },

### ,

### },

a,,### }, {

### ,,

be asinTheorem30. Thenn--1

### E

2a,_,### H

### (I

c_{U}

### +

_{f%)l]z,-}

_{T:r.,ll}

### +

_{2a,,]]z,,}

t=0

### E

2a,,_,### (1

%### +

### 6%)1]z,

### Tz,[

### +

### ,,+

t=0 j--t+l

,=0

and lim,_y, pif andonlyiflim,-_.oe,, 0.

The proofis similarto that of Theorem1andwill therefore beomitted.

Thefollowingestablishesastability theorem for

### Massa’s

iterationprocedure.### THEOREM

32.### ([49,

Theorem### 5]).

Let### T

satisfy the Banachcontractionprinciple. Let x0E### X,

and defineZn+l### SZ,,

whereSsatisfies### (1.8).

Let### {y,,}

C### X,

and defineThen

,=0 j=0 i=0

and lim,,--.oo y,, pif andonlyif lim,..,oo

### ,

0.Weconcludewitharesultonstability for nonexpansivemaps. D. deFigureiredo

### [26,

p.### 230]

establishedthefollowingresult.### THEOREM

33. Let### H

beaHilbert spaceand C a dosed, bounded, and convex subsetin

### H

containing 0. If### T

isany nonexpansiveselfmapof### C,

then,foranyz0 in### C,

the sequence### {z, }

definedbyX

### T:

x,,_,n### 1,2,...and T,,x

### ---Tx,

### "

converges

### strongly

toafixed pointof### T.

### THEOREM

34.### ([19,

Theorem### 2]).

### Let

x0E### X,

where### (X,

### I1"

### II)is

nomedlinear space.### Suppose

thatTisanonexpansive selfmap ofX. Let### ,r(.+l)

n### +

1### f(T, xn)

Xn+ *n+l gn, n 0, 1,2,...where### Tn+lx

_{n+2}

### Tx.

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